Surface defined on the domain?

With this question, I thought that since both the x- and y-values need to be greater than zero for the domain, then obtaining the minimal value would simply mean subbing 0 into the equation. I did that, and got the answer, -1. (Which also matches up as being the correct answer to the question). So, I just basically want to know was what I did to get the answer correct or is there some other way you get the answer? (As, I may be wrong but it just seems that it was a bit of a straightforward and simple approach with getting the answer.)

what you did is correct. it can be explained as follows:
1) the value of z completely depends on the value of x+y.
2)the lower the value of x+y the lower the value of z.
3) the least value of x+y available in D is 0.
so e^0-2 is the answer.

Here the situation was fairly simple so we could use the above reasoning. In general such problems are solved using multivariate calculus.